The basic Jacobian elliptic functions \(\mathrm {sn}(u,k)\) ,
\(\mathrm {cn}(u,k)\)  and \(\operatorname{dn}(u,k)\)  are written as \(\operatorname{sn}{u}\) ,
\(\operatorname{cn}{u}\)  and \(\operatorname{dn}{u}\) , respectively [1]}, [2]}. Here the parameter \(k\) , called
modulus of the functions, is supposed
to be a real number such that \(0<k^2<1\) . These functions
satisfy the relations
\({\rm sn}^2 u+{\rm cn}^{2}u=k^2{\rm sn}^2 u+{\rm dn}^2 u=1\;\Rightarrow \;(k^2-1) {\rm sn}^2 u= {\rm cn}^2 u- {\rm dn}^2 u,\; 1-k^2={\rm dn}^2u-k^2{\rm cn}^2u.\) 
